Researchers investigating the characteristics of gifted children collected data from schools in a large city on a random sample of thirty-six children who were identified as gifted soon after they reached the age of four. The following histogram shows the distribution of the ages (in months) at which these children first counted to 10 successfully. Also provided are some sample statistics.

(a) Are conditions for inference satisfied?

(b) Suppose you read online that children first count to 10 successfully when they are 32 months old, on average. Using a p-value approach, perform a hypothesis test to evaluate if these data provide convincing evidence that the average age at which gifted children first count to 10 successfully is less than the general average of 32 months. Use a significance level of 0.10.

(c) Interpret the p-value in the context of the hypothesis test and the data.

a) Yes, the conditions for inference are satisfied because a random sample of 36 children was taken, the sample is less than 10% of the population, and the shape of the distribution is roughly bell-shaped.

b)Here, we need to perform the hypothesis test using p-value approach with significance level of 0.10.Hypothesis statements:

Null hypothesis:

H0 :

μ ≥ 32

Alternative hypothesis:

H1 :

μ < 32

Here,

μ denotes the population mean age at which gifted children first count to 10 successfully.

So, the test statistic can be calculated as shown below:

tscore=−1.304

where,

sample mean X = 29.8, population standard deviation σ = 5.2, sample size n = 36 and population mean μ0 = 32.

Since the alternative hypothesis is left-tailed, we should use a one-tailed test.

In order to find the p-value for the test statistic, we can use the t-distribution table for degrees of freedom (df) = n-1 = 35 and a significance level of 0.10.Using the table, we get:

p-value = P(T ≤ tscore)= P(T ≤ -1.304)= 0.1016

So, the p-value for the hypothesis test is 0.1016.

c)The p-value is greater than the significance level of 0.10.

Therefore, we do not have enough evidence to reject the null hypothesis.

We can conclude that the data does not provide convincing evidence that the average age at which gifted children first count to 10 successfully is less than the general average of 32 months.

Learn more about random sample from the given link

https://brainly.com/question/13219833

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